I am reading on a book about semiconductor mobility. I have fully understood the definition, but I also noticed that often one talks about high or low mobility. My question is: what are the advantages of having a high mobility? I mean, in the technological applications or experimental uses of the semiconductor/etherojunction.
Do you know any main methods to improve mobility (in Si and/or Ge semiconductors for example)? Any bibliography about them?

2 Answers
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Mobility is a phenomenological parameter used in the context of transistor circuit design. It becomes relevant in different ways depending on the application. As Misha mentioned, you can get higher gain and speed for higher mobility. There are many ways in which mobility has been defined. But from a microscopic point of view the most useful expression is $$\mu=\frac{e\tau}{m^*}$$ where $e$, $\tau$, and $m^*$ are the charge, mean free time between collisions, and effective mass of the carrier. If you look at chapter 2 of Mark Lundtrom's book you will find many ways of computing $\tau$. The methods presented in that book are mostly semiclassical approaches. In other words, it involves partly quantum mechanical and partly classical treatments. $\tau$ is affected by electron-electron, electron-phonon, electron-impurity scattering etc. Looking more into these computations should give you an idea of the different factors that influence mobility. A lot of times mobility simply depends on the quality of fabrication. But there is a theoretical limit which depends only on the material; for example this theoretical limit, for some examples, is $\mu_{\text{graphene}} > \mu_{\text{InSb}} > \mu_{\text{Si}}$ (all at the same temperature). In some other cases (like the High Electron Mobility Transistors (HEMTs) Misha mentioned) you can do some clever engineering to improve mobility. If you read more about HEMTs you'll notice that it falls under the category of reducing electron-impurity scattering.

As far as technological applications go, improving the mobility gives circuit designers the ability to construct circuits with higher performance. In the following paragraphs I will explain what this higher performance exactly means. Before I explain these applications is important to emphasize the difference between the physics and engineering parts of it. There are a lot of different factors that need to be considered while designing a "good" circuit. I will admit that simply improving mobility will not solve all the performance problems. As a matter of fact, certain categories of circuit design have evolved to the point where the performance of the circuit depends mainly on the design and very little on the material parameters. But there are some applications where the physics starts becoming important.

Here is an example where you care about the gain of the transistor. If you are trying to use it as an amplifier you are going to use it in an analog circuit. Higher mobility means higher intrinsic gain of the transistor. The overall gain depends on how these transistors are interconnected. People have come up with many tricks and circuit design strategies to get way more overall gain compared to its intrinsic gain. However, if the intrinsic gain were increased you can get even higher overall gain for the same circuit design.

Another example is a logic circuit where you are interested in using a transistor as a switch. The gate of a Field Effect Transistor (FET) is used to control the flow of current between the drain and source; turning this current "on" and "off" corresponds to the two logic states "0" and "1" (which of them corresponds to "on" and "off" depends on the convention). For example, in an nMOS (or n-MOSFET, where MOS stands for Metal-oxide-semiconductor) device when the gate voltage is above (below) a certain threshold $V_T$ the channel between source and drain becomes conducting (insulating). But this change is not instantaneous. Yes, this time is very short but still finite. In logic circuit language, where you only think about the different logic gates (AND, OR, NOR, XOR, etc.) instead of the actual transistors they are composed of, such individual transistor delays amount to a net "gate delay." Furthermore, these logic gates make up even larger logic modules. These different logic modules work in unison to perform higher-level tasks such as executing code written by a programmer. The reason they are able to work in unison is that their operation is synchronized by the "clock." The period of the clock is determined by the gate delays. The faster the clock the more number of instructions per second the circuit will execute. Well, there are many ways (like parallel processing) of increasing the number of instructions per second; increasing the clock speed is just one of them.

The two examples above demonstrated improvements in gain and speed. There is, in fact, a way to combine these two into a single figure of merit. For a generic amplifier this is called the "Gain–bandwidth product." For transistors one typically uses the term "transition frequency" or "unity-gain bandwidth." As the same suggests, it's the frequency beyond which a transistors stops being an amplifier (gain < 1). There is usually a trade-off between gain and bandwidth. If increase the gain by a certain factor, while keeping your material parameters constant, the bandwidth will reduce by roughly the same factor such that the figure of merit stays constant. I'm not aware of any theoretical expression for the relation between mobility and the figure of merit. But the figure of merit definitely improves if you increase the mobility.

High mobility means 1) better conductivity with the same carrier density and 2) faster response.
One of the methods which is even used in modern processors by Intel is strain. Another trick is used in HEMTs where carriers move in a quantum well and separated from doped layer which provides these carriers. Read wiki atricle.

For the bibliography I'd recommend "Physics of semiconductor devices" by Sze